Proposition 3

Let $\Phi$ be a sesquilinear form for $J$ on $E$, and let $S$ be a basis of $E$. The following conditions are equivalent:

a) The mapping $s_\Phi$ from $E$ into $E^*$ associated with $\Phi$ is bijective.

b) The mapping $d_\Phi$ from $E$ into $E^*$ associated with $\Phi$ is bijective.

c) The element $D_\Phi(S)$ is invertible in $A$.

Indeed condition c) expresses that the matrix of $\Phi$ with respect to $S$ is invertible (Chapter III, § 6, No. 5, Th. 2). Thus c) is equivalent to a) ($§ 1,$ No. 10); similarly c) is equivalent to b).